AN EXTENSION OF THE GENERALIZED PASCAL MATRIX AND ITS ALGEBRAIC PROPERTIES

The extended generalized Pascal matrix can be represented in two diffe rent ways: as a lower triangular matrix Phi(n)[x, y] or as a symmetric Psi(n)[x, y]. These matrices generalize P-n[x], Q(n)[x], and R-n[x], which are defined by Zhang and by Call and Velleman. A product formula for Phi(n)[x, y] has been found which generalizes the result of Call and Velleman. It is shown that not only can Phi(n)[x y] be factorized by special summation, but also Psi(n)[x, y] as Q(n)[xy]Phi(s)(T)[y, 1/ x] or Phi(n)[x, y]P-n(T)[y/x]. Finally, the inverse of Psi(n)[x, y] an d the values of det Phi(n)[x, y], det Phi(n)(-1)[x, y], det Psi(n)[x, y], and det Psi(n)(-1)[x, y] are given. (C) 1998 Elsevier Science Inc.